Abstract
We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere \(C^1\) boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence, positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
This manuscript has no associated data.
References
Agrawal, S., Nahmod, A.: Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner. Nonlinearity 35, 2767–2808, 2022
Bardos, C., Di Plinio, F., Temam, R.: The Euler equations in planar nonsmooth convex domains. J. Math. Anal. Appl. 407, 69–89, 2013
Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York-London (1969)
Di Plinio, F., Temam, R.: Grisvard’s shift theorem near \(L^\infty \) and Yudovich theory on polygonal domains. SIAM J. Math. Anal. 47, 159–178, 2015
Gérard-Varet, D., Lacave, C.: The two-dimensional Euler equations on singular domains. Arch. Ration. Mech. Anal. 209, 131–170, 2013
Gérard-Varet, D., Lacave, C.: The two dimensional Euler equations on singular exterior domains. Arch. Ration. Mech. Anal. 218, 1609–1631, 2015
Han, Z., Zlatoš, A.: Euler equations on general planar domains. Ann. PDE 7, 20–31, 2021
Hölder, E.: Über unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrentzten inkompressiblen Flüssigkeit (German). Math. Z. 37, 727–738, 1933
Kiselev, A., Zlatoš, A.: Blow up for the 2D Euler equation on some bounded domains. J. Differ. Equ. 259, 3490–3494, 2015
Lacave, C.: Uniqueness for two-dimensional incompressible ideal flow on singular domains. SIAM J. Math. Anal. 47, 1615–1664, 2015
Lacave, C., Miot, E., Wang, C.: Uniqueness for the two-dimensional Euler equations on domains with corners. Indiana Univ. Math. J. 63, 1725–1756, 2014
Lacave, C., Zlatoš, A.: The Euler equations in planar domains with corners. Arch. Ration. Mech. Anal. 234, 57–79, 2019
Pommerenke, C.: Boundary Behavior of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992)
Warschawski, S.E., Schober, G.E.: On conformal mapping of certain classes of Jordan domains. Arch. Ration. Mech. Anal. 22, 201–209, 1966
Wolibner, W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long (French). Mat. Z. 37, 698–726, 1933
Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Zh. Vych. Mat. 3, 1032–1066, 1963
Acknowledgements
AZ acknowledges partial support by NSF Grant DMS-1900943 and by a Simons Fellowship.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant conflicts of interest.
Additional information
Communicated by P. Constantin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Han, Z., Zlatoš, A. Uniqueness of Positive Vorticity Solutions to the 2d Euler Equations on Singular Domains. Arch Rational Mech Anal 247, 84 (2023). https://doi.org/10.1007/s00205-023-01908-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-023-01908-2